WASHINGTON—A high-profile investigation into a leak of sensitive information from the Federal Reserve in 2012 has escalated into an insider-trading probe led by a key market surveillance agency and federal prosecutors in Manhattan, according to people familiar with the matter.

But the firm at the center of the probe, Medley Global Advisors, has thrown up a roadblock by claiming a novel defense: It says it is a media organization entitled to special protections under the law, the people said.